Abstract

A discretization of a continuum theory with constraints or conserved quantities is called mimetic if it mirrors the conserved laws or constraints of the continuum theory at the discrete level. Such discretizations have been found useful in continuum mechanics and in electromagnetism. We have recently introduced a new technique for discretizing constrained theories. The technique yields discretizations that are consistent, in the sense that the constraints and evolution equations can be solved simultaneously, but it cannot be considered mimetic since it achieves consistency by determining the Lagrange multipliers. In this paper we would like to show that when applied to general relativity linearized around a Minkowski background the technique yields a discretization that is mimetic in the traditional sense of the word. We show this using the traditional metric variables and also the Ashtekar new variables, but in the latter case we restrict ourselves to the Euclidean case. We also argue that there appear to exist conceptual difficulties to the construction of a mimetic formulation of the full Einstein equations, and suggest that the new discretization scheme can provide an alternative that is nevertheless close in spirit to the traditional mimetic formulations.

Received 30 May 2004Accepted 20 October 2004Published online 09 February 2005

Acknowledgments:

The authors wish to thank Manuel Tiglio for discussions and Luis Lehner and Olivier Sarbach for comments on the paper. This work was supported by Grant No. NSF-PHY0244335, Grant No. NASA-NAG5-13430, DID–USB Grant No. GID-30, Fonacit Grant No. G-2001000712 and funds from the Horace Hearne, Jr., Laboratory for Theoretical Physics and the Abdus Salam International Center for Theoretical Physics.

Article outline:I. INTRODUCTIONII. CONSISTENT DISCRETIZATION OF CONSTRAINED THEORIESIII. LINEARIZED GENERAL RELATIVITY IN TERMS OF METRIC VARIABLESA. Continuum formulationB. DiscretizationC. StabilityIV. LINEARIZED GENERAL RELATIVITY IN TERMS OF ASHTEKAR VARIABLESA. Continuum formulationB. Discretizing the full theory on the latticeC. The linearized theory on the latticeV. DISCUSSION AND CONCLUSIONS